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Thermoelastic damping in bi-layered micro/nanobeam resonators using the dual-phase-lag generalized heat conduction model
Thermoelastic damping is a critical loss mechanism in micromachined resonators used for sensing and communication applications. In this paper, thermoelastic damping of the axisymmetric vibration of laminated circular plate resonators will be discussed. Based on the classical laminated plate theory assumptions, the governing equations of coupled thermoelastic problems are established for axisymmetric out-of-plane vibration of trilayered circular plate with fully clamped boundary conditions. The analytical expression for thermoelastic damping is obtained and the accuracy is verified through comparison with FEM results. Then the effect of material selection and the volume fraction of the covering layers are numerically evaluated. Finally, the thermoelastic damping for different vibration modes is also evaluated.
The current work investigates the transverse vibration of a piezothermoelastic (PTE) nanobeam in the frame of dual-phase-lag thermoelasticity theory. Closed-form analytical expression for the thermoelastic damping (TED) in terms of quality factor for a homogeneous transversely isotropic PTE beam is derived by using Euler–Bernoulli beam theory and complex frequency approach. The size effect of the nanostructured beam is tackled by applying modified couple stress theory (MCST). Detailed analysis on damping of vibration owing to thermal fluctuations and electric potential in the present context under three sets of boundary conditions is attempted to investigate the influences of two-phase-lag parameters, piezoelectric parameter, thermal effect and size-dependent behaviour on energy dissipation caused by TED in PTE beam resonators. Analytical results are illustrated with the help of graphical plots on numerical findings for lead zirconate titanate (PZT-5A) PTE material. The investigation brings out some significant key findings and observations in view of the present heat conduction model.
In light of the certainty of size effect on heat conduction process in extremely small dimensions, the present article seeks to introduce a new theoretical framework for thermoelastic damping (TED) in micro-/nanobeam resonators with circular cross section by utilizing the non-Fourier model of nonlocal dual-phase-lag (NDPL). The first stage involves using NDPL model in order to develop the non-Fourier heat equation of circular cross-sectional beams in polar coordinates. This differential equation can be solved to arrive at temperature distribution in any arbitrary point of the beam. When the constitutive equations of the beam together with the extracted temperature distribution are substituted in the energy-based formulation of TED, an infinite series is produced as the TED relation in the context of NDPL model. Through a comparison study, the reliability of the acquired formula is analyzed. To shed light on how some key factors like nonclassical parameters of NDPL model, beam dimensions and material affect TED, several numerical data are prepared. As per the acquired outcomes, notably at high frequencies of oscillation, the use of NDPL model may profoundly impact the quantity and pattern of TED.
Functionally graded (FG) bi-layer structures have become one of the most promising candidates for micro-devices, which are widely used as high-efficient micro-resonators due to their excellent thermo-mechanical properties. In addition, the design of high performance micro-resonators requires sufficiently accurate analysis of their thermoelastic damping (TED). Nevertheless, the classical analysis model of TED fail on the micro-structures owing to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. To address this issue, present study focuses on investigating the size-dependent TED model of FG bi-layered microbeam resonators for TED analysis by combining the nonlocal dual-phase-lag heat conduction model and the modified coupled stress theory. It is assumed that the FG bi-layered microbeam resonators consist of double FG surfaces. The corresponding governing equation are formulated, and the analytical solution is solved by complex frequency method. The obtained TED model is theoretically verified, and then, the parameter effects of the nonlocal thermal parameter, the material length scale parameter, the power-law index and the vibration modes on the TED are analyzed. This article provides a theoretical analysis model of the TED in FG bi-layered microbeam resonators, which has practical significance in the design of high quality factor devices.
Thermoelastic damping (TED) has been discerned as a definite source of intrinsic energy loss in miniaturized mechanical elements. The size-dependent structural and thermal behavior of these small-sized structures has been proven through experimental observations. As a first attempt, this article exploits nonlocal strain gradient theory (NSGT) and nonlocal dual-phase-lag (NDPL) heat conduction model simultaneously to acquire a mathematical formulation and analytical solution for TED in nanobeams that can accommodate size effect into both structural and heat transfer fields. For this purpose, the coupled equations of motion and heat conduction are firstly extracted via NSGT and NDPL model. Next, by deriving the distribution of temperature from heat conduction equation and substituting it in the motion equation, the unconventional thermoelastic frequency equation is established. By deriving the real and imaginary parts of the frequency from this equation and employing the definition of quality factor, an explicit solution is given for approximating TED value. The veracity of the proposed model is checked by comparing it with the solutions reported in the literature for specific and simpler cases. A diverse set of numerical results is then presented to appraise the influence of some factors like structural and thermal nonlocal parameters, strain gradient length scale parameter, geometrical parameters, mode number and material on the amount of TED. According to the results, use of NDPL model yields a smaller value for TED than DPL model, but prediction of NSGT about the magnitude of TED, in addition to the relative amounts of its two scale parameters, strongly depend on other factors such as aspect ratio, vibration mode and material type.
Functionally gradient materials (FGM) in nanobeams are interesting issues in the theory of elasticity and thermoelasticity regarding thermal and mechanical stress. These advanced heat-resistant materials are used as structural components in contemporary technology. The thermoelastic interactions in functionally graded nanobeams (FGN) have been studied in this article. The basic equations that control the introduced model have been established based on the Euler–Bernoulli beam concept, Eringen’s theory, and the two phase-lag fractional heat conduction model. The heat equation has been modeled and fractionalized into a new formula that includes nonsingular and nonlocal differential operators. The physical properties of the nanobeam vary in graded according to its thickness. The FGN nanobeam is subject to a time-dependent and periodically varying heat flow. The differential equations are analyzed analytically in the Laplace transform field. The responses in the nanobeam are graphically depicted for various fractional-order values, the influence of the nonlocal parameter and the periodic frequency of the heat flux. The results show that the gap between classical and nonlocal theories widens with increasing nonlocal parameters and decreasing nanobeam length.
Many challenges in different applied fields of research, such as materials science, viscoelasticity, biological sciences, physics, and mechanical engineering, require the study of derivative operators using single singular or nonsingular kernels. Atangana and Baleanu (AB) constructed a novel fractional derivative without a singular kernel based on the extended Mittag–Leffler function to overcome the singular kernel problem seen in previous definitions of fractional‐order derivatives. In this article, we provide a novel mathematical thermoelastic heat conduction model that includes the fractional AB derivative operators. In addition, the Moore–Gibson–Thompson (MGT) equation has been incorporated into the proposed heat transport model. The proposed model has been applied to study an infinite orthotropic material with a cylindrical aperture, and the thermal conductivity coefficient of the body depends on the temperature change. The Laplace transform approach has been used to solve the system of governing partial differential equations (PDEs). To assess the validity of the proposed model and for the purposes of comparison, the numerical results have been depicted in figures as well as in tables.
The current study aims to introduce a new generalized photothermal model in which heat equation is described based on the Moore–Gibson–Thompson (MGT) equation. The thermo-optical transition process can be understood, and the interaction between elastic plasma waves and heat can be investigated and explained using the suggested model. The proposed model was used to investigate the thermal and photoacoustic effects in an infinitely constrained solid cylinder of semiconductor material that was crossed into a fixed magnetic field and subjected to a high-intensity laser heal flux. The Laplace transform technique is used to derive the numerical expressions for the components of thermal stresses, displacement, temperature field, and carrier density. The propagation of thermal, elastic, and plasma waves, as well as the distributions of each studied field, was investigated and described. The comparison is also used to evaluate the impact of thermoelastic response characteristics such as thermal relaxations, temperature frequency, and lifetime on the photo-thermoelastic response.
Thermal and mass diffusion processes are important issues in a variety of engineering applications and scientific disciplines. The main objective of this research is to develop a new model that demonstrates diffusion in thermoelastic solids and compares the strain/temperature fields and mass diffusion. The proposed model is an extension of the Quintanilla model [1]. In the new model, Fourier’s and Fick’s laws have been improved by including the relaxation times in the Green-Naghdi theory in the framework of Moore-Gibson-Thompson (MGT) heat equation. Based on the introduced model, a one-dimensional half-space problem is considered. The surface surrounding the half-space is exposed to chemical potential and thermal shocks. Our findings indicate that the considered physical fields have a non-zero value only in a limited area and disappear outside this area. This result fully demonstrates the validity of the proposed model because the nature of velocities is limited by heat and diffusive waves.
In the current article, in the presence of thermal and diffusion processes, the equations governing elastic materials through thermodiffusion are obtained. The Moore–Gibson–Thompson (MGT) equation modifies and defines the equations for thermal conduction and mass diffusion that occur in solids. This modification is based on adding heat and diffusion relaxation times in the Green–Naghdi Type III (GN-III) models. In an unbounded medium with a cylindrical hole, the built model has been applied to examine the influence of the coupling between temperature and mass diffusion and responses. At constant concentration as well as intermittent and decaying varying heat, the surrounding cavity surface is traction-free and is filled slowly. Laplace transform and Laplace inversion techniques are applied to obtain the solutions of the studied field variables. In order to explore thermal diffusion analysis and find closed solutions, a suitable numerical approximation technique has been used. Comparisons are made between the results obtained with the results of the corresponding previous models. Additionally, to explain and realize the presented model, tables and figures for various physical fields are presented.
In this note we propose the Moore-Gibson-Thompson heat conduction equation with two temperatures and prove the well posedness and the exponential decay of the solutions under suitable conditions on the constitutive parameters. Later we consider the extension to the Moore-Gibson-Thompson thermoelasticity with two temperatures and prove that we cannot expect for the exponential stability even in the one-dimensional case. This last result contrasts with the one obtained for the Moore-Gibson-Thompson thermoelasticity where the exponential decay was obtained. However we prove the polynomial decay of the solutions. The paper concludes by giving the main ideas to extend the theory for inhomogeneous and anisotropic materials.
This paper presents a theoretical investigation on the response of free vibration of functionally graded material (FGM) micro-plates with thermoelastic damping (TED). Continuous through thickness variation of the mechanical and thermal properties of the FGM plate is considered. By employing the simplified one-way coupled heat conduction equation and Kirchhoff’s plate theory, governing equations for the free vibration of the FGM micro-plates with thermoelastic coupling effect are established, in which stretching-bending coupling produced by the material inhomogeneity in the thickness direction is also considered. The heat conduction equation with variable coefficients is solved effectively by a layer-wise homogenization approach. Harmonic responses of the FGM micro-plates with complex frequency are obtained from the mathematical similarity between the eigenvalue problems of the FGM micro-plate with TED and that of the homogenous one without TED. The presented analytical solutions are suitable for evaluating TED in FGM micro-plates with arbitrary through-thickness material gradient, geometry and boundary conditions. Numerical results of TED for a ceramic-metal composite FGM micro-plate with power-law material gradient profile are illustrated to quantitatively show the effects of the material gradient index, the plate thickness, and the boundary conditions on the TED. The results indicate that by adjusting the physical and geometrical parameters of the FGM micro- plate, one can get the minimum of the TED which is even smaller than that of the pure ceramic resonator.
This paper aims to present an explicit relation for thermoelastic damping in nanobeams capturing the small-scale effects on both the continuum mechanics and heat conduction domains. To incorporate small-scale effects, the coupled equations of motion and heat conduction are obtained by employing the nonlocal elasticity theory and the dual-phase-lag heat conduction model. Adopting simple harmonic forms for transverse deflection and temperature increment and solving the governing equations, real and imaginary parts of the frequency are extracted. According to the complex frequency approach, a closed-form size-dependent expression for evaluating thermoelastic damping in nanobeams is derived. To clarify the influence of nonlocality and dual-phase-lagging on the amount of thermoelastic damping, numerical results are compared with the ones predicted in the framework of classical continuum and heat conduction theories. Findings reveal that the size effect on both the continuum mechanics and heat conduction modeling of nanobeams is not negligible. A number of parametric studies are also conducted to indicate the effect of beam dimensions, boundary conditions and type of material on the value of thermoelastic damping.
Under the condition that microresonators work at room temperature or vaccum, thermoelastic damping is one of the main mechanisms of energy dissipation. Thermoelastic damping caused by the internal consumption of thermoelastic materials has always prevented the improvement of the quality of microresonators. In this paper, the theoretical model of thermoelastic damping in fully clamped bilayered plate microresonators based on the theory of three-dimensional heat conduction is first established and then verified to be equivalent to the previous single-layer model or not through the formula derivation. Analysis on thermoelastic damping at the first-order frequency where microresonators usually work is carried out afterwards. The differences of thermoelastic damping in the present three-dimensional model with different materials are investigated, including the convergence speed and the value of thermoelastic damping with different thicknesses. Then, with different lengths, widths, and thicknesses, but the same combination of materials, the thermoelastic damping is investigated in the present model. Furthermore, the present bilayered model is compared with the single-layer model to investigate their equivalent relationship. Finally, the present three-dimensional model is compared with the one-dimensional model and FEM models to investigate its feasibility.
Thermoelastic damping (TED) can lead to energy loss in microscale resonators, which is an intrinsic mechanism. To minimize the energy loss, it is required to determine the TED of resonators. Laminated plate resonators are commonly used in practice. However, existing researches on TED of the laminated resonators use mainly the one-dimensional (1D) heat conduction model, as the 3D governing equation is complicated, which cannot show the influences of boundary conditions along the supporting edges. In this paper, the governing equation of thermoelastic problems with 3D heat conduction was established for the out-of-plane vibration of the laminated rectangular plate. The analytical expression of the TED was derived using its physical meaning, namely, the ratio of the energy dissipated to the total elastic strain energy stored per cycle of vibration. It was found that the size and shape of the plate affect crucially the TED. The values of TED for higher-order vibration modes were also evaluated. Most importantly, the influences of supporting conditions and heat conduction conditions along the four edges were studied, which is the first report for laminated plates. The present approach can provide guidance for the design of high-quality bilayered resonators.
Analysis and numerical results are presented for the thermoelastic dissipation of a homogeneous isotropic, thermally conducting, Kelvin–Voigt type circular micro-plate based on Kirchhoff’s Love plate theory utilizing generalized viscothermoelasticity theory of dual-phase-lagging model. The analytical expressions for thermoelastic damping of vibration and frequency shift are obtained for generalized dual-phase-lagging model and coupled viscothermoelastic plates. The scaled thermoelastic damping has been illustrated in case of circular plate and axisymmetric circular plate for fixed aspect ratio for clamped and simply supported boundary conditions. It is observed that the damping of vibrations significantly depend on time delay and mechanical relaxation times in addition to thermo-mechanical coupling in circular plate under resonance conditions and plate dimensions.
In this paper, the thermoelastic damping of the in-extensional vibration of rings is analyzed by dual-phase-lagging generalized thermoelasticity theory. In this work, the Q-factor for thermoelastic damping is derived. The effect of the ring radius and thickness, the rotating speed, mode numbers and ambient temperatures are discussed. This work represented that the lengths of the ring, the speed of the rotation and the relaxation times of the dual-phase-lagging have significant effects on the thermo-mechanical damping parameter.
Thermoelastic damping (TED) has been recognized as a significant mechanism of energy loss in vacuum-operated microresonators. Three-layered microbeams are common elements in many microresonators. However, only the model for TED in the three-layered microbeams with symmetric structure has been developed in the past. The first and the third layers in these beams have the same thickness and material properties. Thus, the temperature field is symmetric in these beams. In this paper, an analytical expression for TED in the asymmetric three-layered microbeams is developed in the form of an infinite series. The temperature fields in the asymmetric three-layered microbeams are asymmetric. The total damping is obtained by computing the energy dissipated in each layer. It is seen that the values for TED computed by the present model agree well with those computed by the finite-element model. The limitations of the present model are assessed. A simple model is also presented by retaining only the first term. The accuracy of the simple model is also discussed. The present model can be used to optimize the design of three-layered microbeams.
Understanding the behavior of rotating materials and structures on small scales is crucial for many scientific and engineering fields, and such studies play an important role in this regard. This paper aims to propose a novel paradigm for analyzing the vibrational characteristics of thermoelastic nanobeams with diverse physical attributes. The incorporation of size effects in the structural and thermal constitutive relationships involves the consideration of nonlocal elasticity theory (NET) and the modified couple stress (MCS) model, together with the utilization of the Euler–Bernoulli assumptions for thin beams. The work also involves the development of a new non-Fourier thermoelasticity model that incorporates the Moore–Gibson–Thompson (MGT) equation. Furthermore, it was taken into account that the thermal conductivity of the flexible materials is not consistent but rather changes with temperature. Periodic pulse heating was applied to rotating nanobeams, and the behavior of the nanobeams was investigated with respect to thermal, rotational, and length-scale effects. To demonstrate the impact of the distinctive characteristics of the MCS and MGT thermoelastic models on the physical fields, a range of numerical data are presented. The study also investigated the propagation characteristics of thermo-mechanical waves, taking into account aspects such as thermal relaxation time and the influence of temperature change on physical properties. Based on the observed results, including the size impact in the structural and thermal equations can lead to significant disparities when compared to conventional models. The inclusion of the length-scale component in the MCS theory, which increases the rigidity and hardiness of the nanobeam structure, may help to explain the observed effect.
Functionally graded sandwich micro/nano-structures have attracted great attention due to the capability to resist high noise and thermal stress in a non-isothermal environment. Additionally, the design of high quality-factor micro/nano-resonators requires accurate estimation of their thermoelastic damping. However, the classical thermoelastic damping models fail at the micro/nano-scale due to the influences of the size-dependent effects related to heat transfer and elastic deformation. This work aims to investigate the influences of the size-dependent effects on the thermoelastic damping of functionally graded sandwich micro-beam resonators by combining the nonlocal dual-phase-lag heat conduction model and the nonlocal elasticity model. It is assumed that the functionally graded sandwich micro-beam resonators consist of a ceramic core and functionally graded surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of thermoelastic damping is obtained by complex frequency method. Numerical results are analyzed for the effects of the thermal nonlocal parameter, the elastic nonlocal parameter, the power-law index, and the vibration modes on the thermoelastic damping of functionally graded sandwich micro-beam resonators. The results show that the thermoelastic damping of functionally graded sandwich micro-beam resonators can be adjusted by the suitably modified parameters, which strongly depends on the double nonlocal effects and the power-law index.
In this research, a new fractional framework for the non-Fourier thermal conductivity theory has been developed and analyzed in detail. Furthermore, a dual-phase-lag (DPL) model is presented to consider the discrete time intervals necessary for various microstructural processes. To overcome the singularity problem associated with traditional fractional derivatives, the Mittag-Leffler function is selected as a complementary kernel alongside the power-law function that defines the time-fractional derivative. The proposed model is based on a new fractional derivative that incorporates the Atangana and Baleanu operators with a two-parameter Mittag-Leffler kernel. The fractional DPL heat transfer model is employed to investigate transient heat transfer in functionally graded thermoelastic nanobeams. In addition, through the amalgamation of the Hamiltonian principle, non-local elasticity, and the Euler–Bernoulli beam theory, a set of mathematical equations has been formulated to delineate the characteristics of FG nanoscale beams. The Laplace transform method has been employed to ascertain the responses of thermo-mechanical fields in the transformed domain. To validate the proposed approach, a numerical illustration and a graphical depiction of pertinent numerical data have been included. The thermomechanical characteristics of nanobeams have been thoroughly investigated, encompassing their responsiveness to the power law index, nanoscale parameters, and fractional operators. Lastly, comparisons between different alternate kernels have revealed that both the fractional DPL models and the presented solutions effectively capture the heat transfer across the medium.
Thermoelastic damping (TED) is one of the key factors for lowering the quality factor (Q-factor) of micro/nano-resonators. However, due to a complex small-scale effect and governing equations of non-homogeneous isotropic materials, the existing TED models usually focus on homogeneous isotropic micro/nano-resonators. In this paper, a closed-form TED model is derived to estimate the influence of the small-scale effect on TED of transversely isotropic micro/nano-resonators. The surface effect and the dual-phase-lag model are included to distinguish the influence of mechanical and thermal small-scale effects on TED, respectively. The obtained TED model is theoretically verified. The results indicate that TED values are underestimated if the classical TED model is employed. Moreover, a critical thickness can be determined by the frequency shift curve is proposed. The small-scale effect results in higher TED values within the critical thickness. However, since the small-scale effect has a weak influence on TED, it can be neglected when the resonator thickness is higher than the critical thickness. Additionally, the surface effect plays a dominant role in improving the TED of nano-resonators. In this paper, a more reasonable theoretical approach for estimating TED in transversely isotropic micro/nano-resonators is provided.
The quality factor is a key parameter of micro-electro-mechanical system resonators, and the internal friction of structural materials in the micro resonator caused by thermoelastic damping has been limiting the improvement of the quality factor. In this paper, thermoelastic damping models of trilayered composite microplates with three kinds of boundary conditions are studied. Then the convergence of thermoelastic damping with different material combinations is analyzed, and that with three kinds of boundary conditions is also analyzed. Next, the finite element model is compared with the analytical model, and the accuracy of the analytical model is verified. Finally, the effects of the coating material and the coating thickness on thermoelastic damping of micro resonators are analyzed. The results show that the Zener modulus of the coating material has a great influence on the convergence of the analytical model; the boundary condition has little effect on the convergence of the analytical model; the maximum thermoelastic damping of composite plates is related to Zener modulus of coating; the maximum thermoelastic damping of composite plates is related to the thickness of coating.
Precisely calculating quality factor based on thermoelastic damping (TED) is of great importance for the design and optimization of micro/nano-resonators. With the consideration of size-dependent effects in the thermal and mechanical fields and the non-Fourier (NF) heat conduction effect, analytical expressions of TED in the micro/nanobeam resonator are proposed by adopting the theories of the modified-couple-stress (MCS) model and the nonlocal dual-phase-lag (DPL) model. TED models expressed in the series form and the explicit form are both developed basing on the energy-definition approach. The differences among TED models are discussed. In the simulation, silicon and gold, which are two representative materials with typical intrinsic length-scale quantities and phase-lag times, are selected for illustration. Additionally, the nonlocal thermal length-scale parameter refers to the mean-free-path (MFP) of energy carriers, and the MCS mechanical length-scale parameter is not constant but proportional to the beam thickness. TED results obtained by the classical model and the previous DPL model are also shown for comparison. The peak phenomena of TED spectra including the peak damping and the peak frequency are investigated according to the present simple model. Results reveal that TED in micro/nanobeam resonators is of dependence observably on the effect of nonlocal DPL-NF heat conduction associated with the thermal field, and the quality factor is improved by the MCS mechanical size-dependent effect.
Thermoelastic damping (TED) in a simply supported functionally graded material (FGM) micro rectangular plate is investigated analytically based on the higher-order shear deformation plate theory. The equations of motion for the thermo-elastic coupled transverse free vibration and the heat conduction equation coupled are derived based on the Levinson plate theory and the one-way coupled heat conduction theory. A semi-analytical solution of the heat conduction equation with variable coefficients is obtained by using a layer-wise homogenization approach. The complex frequency of the micro plate including TED is determined in terms of the frequency of the corresponding isothermal homogenous Kirchhoff plate by using the mathematical similarity between the eigenvalue problems for the two types of plates. The effects of the shear deformation, the material gradient and the geometry on the TED are examined in detail for the FGM micro plate made of ceramic–metal constituents with the power-law gradient profile. The numerical results show that the TED evaluated by the Levinson plate theory is smaller than that by the Kirchhoff plate theory. Therefore, for a thick or moderately thick micro plate the Levinson plate theory can provide a more accurate prediction of the TED than the Kirchhoff plate theory.
This paper assesses thermoelastic damping (TED) in circular nanoplates by incorporation of the small-scale effect into structural and thermal domains. The nonlocal elasticity theory and dual-phase-lag (DPL) heat conduction model are exploited for achieving the size-dependent coupled thermoelastic equations. By choosing time-harmonic and asymmetric form for deflection and temperature change, and solving the size-dependent thermoelastic eigenvalue problem, the damped natural frequency of circular nanoplate is extracted. On the basis of the complex frequency approach, an analytical relation, for the description of TED in circular nanoplates, is derived. For different boundary conditions and vibration modes, a comparison study is performed between the size-dependent results and those provided by classical continuum mechanics and heat conduction theories. Outcomes show a discrepancy between results acquired by the size effect on the structure and heat conduction. It is elucidated that how nonlocal and DPL characteristic parameters can represent the size-dependent phenomenon and can affect the magnitude of TED. A comprehensive parametric study is conducted to specify the influence of boundary constraints, vibration mode and the type of material on the amount of energy loss from TED.
Thermoelastic damping (TED), as a main source of intrinsic energy dissipation, is crucial to the design of the micro/nano-devices and -systems with higher quality factor (Q-factor). However, the classical analysis model of TED fails in micro/nano-scale due to the influence of small-scale effect. The present article focuses on investigating the influence of scale effect on the TED of micro-beam resonators by considering stress nonlocal and higher-order strain gradient effects. Firstly, the governing differential equations are formulated by employing the nonlocal strain gradient theory (NSG) in conjunction with the dual-phase-lag (DPL) heat conduction model. According to the assumption of vibration mode and the boundary conditions, the size-dependent Q-factor expression of TED is derived by the complex frequency method. Finally, the influences of various parameters on the TED of micro-beam resonators, such as nonlocal parameter, length scale coefficient, slenderness ratio and material type, are discussed in detail. And then, the analysis results are compared with the TED of classical thermal-mechanical model. This article provides a novel theoretical analysis model of the TED in micro/nano-meter scale field, which has practical significance in the design of high-efficiency devices and systems.
Microresonators used for microelectromechanical systems (MEMS) are becoming more and more complex in terms of structural geometry and mode shapes. Accompanying this increase in complexity is the challenge to develop an accurate analytical method to evaluate the thermoelastic damping (TED). This paper firstly presents a generalized methodology for TED in axisymmetric vibration of circular plate microresonators covered by multiple partial coatings. The frequency and mode shape function of plates are solved by the Kirchhoff–Love plate theory. The thermoelastic temperature fields are acquired by solving the one-way coupled heat conduction along the thickness direction with Green’s function method, accordingly obtaining the effects of partial coating on the microplate. The present model can reduce to that of the fully covered bilayer plates, and is validated by finite element model (FEM). This methodology produces an explicit model for TED in the form of one infinite series with rapid converge, and a simple model is given by retaining the first term. The accuracy and practicality of the simple model are investigated. The results suggest that SiO2 coating with thermally perfect interfaces can reduce the TED at the fundamental mode. However, SiO2 coating significantly increases TED values with thermally imperfect interfaces. Three distinctive peaks of TED spectra in Si/DLC/SiO2 plates are observed. This work can be utilized to optimize the TED in the composite circular microplates and gain more insights into the intrinsic dissipation.
Thermoelastic damping (TED) has been proved as an intrinsic mechanism of energy dissipation in the microelectromechanical systems (MEMS) resonators. However, the previous TED models developed for the fully covered bilayer beam resonators cannot be used for the partially covered cases. This paper firstly derives an analytical TED model for partially covered bilayer microbeams. The bilayer beam performs small-amplitude vibration in pure bending mode, and the mode shape is achieved from the dynamic of Euler-Bernoulli beam. To obtain the coupled temperature field, Green's functions are utilized to solve the heat conduction along thickness and length directions within the framework of Fourier's law. The expression for TED is derived in the form of an infinite series. The present TED model can reduce to that of a fully covered bilayer beam, and matches well with the finite element method (FEM). The behaviors of TED spectrum are investigated comprehensively. Two comparable Debye peaks are noticed at approximately two corresponding critical frequencies. The partial coating greatly reduces the peak values at high critical frequency, but causes an additional TED peak at low critical frequency. The TED peaks of the coating slightly increase as the length increases. The effects of the length and position of the metal coating on the TED at the fundamental frequency are significant. To reduce TED, the metal coating should be located away from the substrate clamped end. This paper provides a developing methodology for controlling TED.
Considering the heat-conduction dimension (HCD) and adopting the dual-phase-lagging (DPL) non-Fourier theory, analytical models of thermoelastic damping (TED) and frequency shift for the rectangular cross-section micro/nano-ring resonators are first derived in the series form in this work. In the modeling procedure, one of emphases is the estimation and solution of the governing equation of coupled thermoelasticity considering one-dimensional (1D) and two-dimensional (2D) heat conduction. The orthogonality-integration method of the trial function is used to solve the temperature profile functions. The TED expressions obtained by the energy-definition approach and the complex-frequency approach are both demonstrated. The previous models are compared with the present proposed models. The influences of the dual-phase-lagging non-Fourier (DPL-NF) effect, HCD, the material selection, and the ratio of dual-phase-lagging times on TED are investigated. The dependences of TED and the frequency shift on the geometrical parameters involving the mean radius and radial depth of the ring, and the modal order are also examined. The results show that TED spectra and frequency shift are significantly affected by the HCD and the DPL-NF effect.
Elastothermodynamic damping is one of the most important energy loss mechanisms in micro/nano mechanical resonators, which is related to the high sensitivity and high resonance frequency characteristics of resonators. With the wide application of resonators, multilayer microplate resonators have been applied. In this paper, the elastothermodynamic damping models of three-layer of Kirchhoff–Love microplate under three typical boundary support conditions of Clamped-Clamped-Clamped-Clamped (C-C-C-C), Clamped-Free-Clamped-Free (C-F-C-F) and Clamped-Free-Free-Free (C-F-F-F) are established. Firstly, the thermoelastic mechanical equations of the microplate are given, and then the three-dimensional temperature distribution functions of the microplate are obtained based on the generalized orthogonal function method. Finally, the elastothermodynamic damping models are established based on the composite multilayer plate theory. The model in this paper is compared with the previous model and the finite element results.
Thermoelastic damping (TED) is one of the main internal energy dissipation mechanisms in micro-/nano-resonators. Accurate evaluation of TED is important in the design of micro-electromechanical systems and nano-electromechanical systems. In this paper, a theoretical analysis on the TED in functionally graded material (FGM) micro-beam resonators is presented. Equations of motion and the heat conduction equation governing the thermodynamic coupling free vibration of non-homogenous micro-beams are established based on the Euler–Bernoulli beam theory associated with the modified couple stress theory. Material properties of the FGM micro-beam are assumed to change in the depth direction as power-law functions. The layer-wise homogenization method is used for solving the heat conduction equation. By using the mathematical similarity of eigenvalue problem between the FGM beam and the reference homogeneous one, the complex natural frequency including TED is expressed in terms of the natural frequency of the isothermal homogenous beam. In the presented numerical results, influences of various characteristic parameters, such as beam thickness, material gradient index, structure size, vibration mode and boundary conditions, on TED are examined in detail. It shows that TED decreases with the increases in the values of length scale parameters because the latter lead to the increase in structural stiffness.
Thermoelastic damping (TED) affects the quality factors of vacuum-operated micro/nanobeam resonators significantly. In this work, by adopting the non-Fourier theory of dual-phase-lag (DPL) model, an analytical formula of TED in micro/nanobeam resonators with circular cross-section is first developed. Moreover, for micro/nanobeam resonators with rectangular cross-section, the series-form type of DPL-TED model is also proposed and compared with the modified existing model. The characteristics of TED spectra with the single-peak, dual-peak, and multiple-peak phenomena are explored. The simulation results reveal that the ratio of dual-phase-lag times and the characteristic dimension of beams such as the radius and thickness have significant influences on TED behaviors. In addition, temperature distributions in micro/nanobeams exhibit an apparent distinction under the DPL non-Fourier effect.
We consider a thermoelastic theory where the heat conduction is described by the Moore–Gibson–Thompson equation. In fact, this equation can be obtained after the introduction of a relaxation parameter in the Green–Naghdi type III model. We analyse the one- and three-dimensional cases. In three dimensions, we obtain the well-posedness and the stability of solutions. In one dimension, we obtain the exponential decay and the instability of the solutions depending on the conditions over the system of constitutive parameters. We also propose possible extensions for these theories.
In this article, the size-dependent behavior of micro-beams with the thermoelastic damping (TED) phenomenon is studied. The coupled thermoelasticity equations are derived on the basis of the modified couple stress theory (MCST) and dual-phase-lag (DPL) heat conduction model. By solving these coupled equations simultaneously, a closed-form expression for the TED parameter in micro-beams is presented which considers the small-scale effects incorporation. Then, the effect of various parameters on TED in micro-beams, such as micro-beam height, the type of material, boundary conditions, and aspect ratio is investigated. The results show that the influence of utilizing non-classical continuum and thermoelasticity theories on the amount of TED and the critical thickness is significant in small scales.
In this article, thermoelastic damping (TED) in functionally graded material (FGM) circular micro plates is analyzed based on classical plate theory and one-way coupled heat conduction equation. The material properties are assumed to be varied continuously in the thickness direction of the plate. A one-way coupled heat conduction equation with variable coefficients is solved by using a layer-wise homogenization approach. The complex frequency, including TED, of the FGM micro plate is obtained in terms of the natural frequency of the corresponding isothermal homogenous plate without TED from the mathematical similarity between the eigenvalue problems of governing differential equations of the two kind structures. Numerical results of TED for a ceramic-metal composite FGM circular micro plate are presented quantitatively to show the effects of the material gradient index, the geometry, the vibration mode shapes, and the environmental temperature on the TED in detail. The present mathematical model and solution method can be also used to evaluate TED in the FGM circular and annular plates with other through-thickness material gradient forms.
The design of laminated composite microplate resonators with high quality (Q) factors requires careful analysis of thermoelastic damping since it is an inherent intrinsic energy dissipation mechanism. This paper presents a temperature field in a general trilayered microplate with thermally perfect interfaces based on the integral transform approach. In addition, an analytical model to calculate thermoelastic damping in general trilayered fully clamped microplates is developed. Total thermoelastic damping for trilayered microplates can be expressed as a sum of the normalized energy dissipated in each layer. In order to validate the present model, the results calculated by the present model are compared with those calculated by finite element method (FEM). Our results show that the energy dissipation in the middle part is less than that in the outer parts for homogenous microplates. Thermoelastic damping peaks in trilayered microplates are discussed, which are associated with the critical damping frequency and the Zener's modulus of each layer. When the Zener's modulus of one layer is 2–3 orders of magnitude higher than that of another layer and the critical damping frequency of this layer is also 2–3 orders of magnitude higher than that of another layer, the thermoelastic dissipation spectrum will exhibit multiple peaks. It is also observed that the second layer may exhibit negative energy dissipation in SiC/Ti/Au and Si3N4/Si/Au trilayered microplates.
To minimize thermoelastic damping (TED) loss of energy, it is of great significance to determine TED for resonators. In this paper, we established the governing equation of thermoelastic problems with two-dimensional heat conduction for the out-of-plane vibration of bilayerd circular plate and derived the analytical express of TED. The same problem is simulated by using finite element method (FEM). The analytical results obtained by the present 2-D model show a good agreement with the results obtained by FEM. It is found that the radius and the thickness of the plate play important roles on TED. The values of TED for higher-order vibration modes were also evaluated. Finally, the influence of imperfect thermal contact on the interface was studied and two peak values were found.
In this paper, thermoelastic damping (TED) in free vibrating functionally graded material (FGM) micro beams with rectangular cross sections is investigated. The material properties of the micro beams are assumed to be varied continuously in the thickness direction. Based on the classical beam theory and coupled thermo-elastic dynamics, governing equations coupled with the thermal effects derived in terms of the deflection and temperature rise filed. A layer-wise homogenization method is proposed to solve the one-way coupled heat conduction equation with variable coefficients. By using the mathematical similarity between the vibration equation of the FGM beam and that of the reference homogenous beam, complex frequency of the FGM beam including TED is expressed in the terms of that of the reference homogenous beam without TED. Numerical results of TED are obtained for the specific material constituents and the power law gradient profile. The effects of the material gradient index, beam thickness, vibration modes and boundary conditions on the TED of the FGM micro beams are studied in detail.
This article deals with the thermoelastic damping problem in a functionally graded (FG) Timoshenko microbeam. Thermal and mechanical properties of the microbeam vary in the thickness direction according to the power law relation. Employing Timoshenko beam theory, the governing dynamic equation coupled with thermal effects of the FG microbeam is developed. Afterwards, Using the Taylor series expansion for material properties, the heat conduction equation is solved analytically for temperature in the form of a power series. The free vibration of the FG microbeam is analyzed to achieve the natural frequencies and thermal damping ratio of the FG microbeam. The effect of FG index on the thermoelastic damping ratio is investigated in different aspect ratios. Also comparison studies are made between the results obtained from the models based on the Euler–Bernoulli and Timoshenko beam theories.
Accurate determination of thermoelastic damping (TED) is very challenging in the design of microresonators with composite structures. This paper investigates TED in the bilayered microplate. The temperature field in the bilayered microplate with a thermally perfect interface subjected to a time-harmonic excitation is presented. The total damping is obtained by computing the energy dissipated in each layer. An analytical model in the form of an infinite series for TED in the bilayered fully clamped rectangular and circular microplates is developed, and the convergence rate of the present model is studied. For TED in the bilayered cantilever and fixed-fixed microplates oscillating at the first natural frequency, an approximate analytical model is also presented based on Rayleigh's method. TED calculated by the present model agrees well with that obtained by the finite element method (FEM). Present results show that the effects of the thickness and the property of the metallic layer on the peak damping are significantly. The present model can be used to optimize the design of the high-Q bilayered microplate resonators